nLab supergeometry

Idea

Supergeometry is the (higher) geometry over the base topos on superpoints modeled on the canonical line object $\mathbb{R}$ in there.

As ordinary differential geometry studies spaces – smooth manifolds – that locally look like vector spaces, supergeometry studies spaces – supermanifolds – that locally look like super vector spaces.

As ordinary algebraic geometry studies spaces – schemes – that locally look like affine spaces, supergeometry studies superschemes.

From the point of view of noncommutative geometry, the supergeometry is a very mild special case of noncommutativity in geometry: some coordinates commute, some anticommute.

For more see at geometry of physics – supergeometry.

Supergeometry may defined over an arbitrary commutative ring (Schwarz and Shapiro 06).

Related concepts

• signs in supergeometry

• superpoint

• superspace

  • supermanifold, super scheme

  • spectral super-scheme

• supersymmetry

• super-spacetime, super-Minkowski spacetime, super Poincare group

• super-vielbein

• supergravity

• Super Gerbes

• bosonic modality

• super-Cartan geometry

  • superconformal geometry

geometries of physics

$\phantom{A}$(higher) geometry$\phantom{A}$	$\phantom{A}$site$\phantom{A}$	$\phantom{A}$sheaf topos$\phantom{A}$	$\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$
$\phantom{A}$discrete geometry$\phantom{A}$	$\phantom{A}$Point$\phantom{A}$	$\phantom{A}$Set$\phantom{A}$	$\phantom{A}$Discrete∞Grpd$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$CartSp$\phantom{A}$	$\phantom{A}$SmoothSet$\phantom{A}$	$\phantom{A}$Smooth∞Grpd$\phantom{A}$
$\phantom{A}$formal geometry$\phantom{A}$	$\phantom{A}$FormalCartSp$\phantom{A}$	$\phantom{A}$FormalSmoothSet$\phantom{A}$	$\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$SuperFormalCartSp$\phantom{A}$	$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$	$\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$

$\,$

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$	$\phantom{A}$fin. gen.$\phantom{A}$ $\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

References

Some historically influential general considerations are in

• Yuri Manin, New Dimensions in Geometry, talk at Arbeitstagung, Bonn 1984

Introductory lecture notes include

• Yuri Manin, chapter 4 of Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften 289, Springer 1988

• Daniel Freed, Five lectures on supersymmetry

• Gennadi Sardanashvily, Lectures on supergeometry (arXiv:0910.0092)

Discussion via functorial geometry:

• Claudio Carmeli, Lauren Caston, Rita Fioresi, Mathematical Foundations of Supersymmetry, EMS Series of Lectures in Mathematics Volume: 15 (2011) (ISBN:978-3-03719-097-5, arXiv:0710.5742)

Discussion of traditional algebraic geometry for super-schemes:

• Ugo Bruzzo, Daniel Hernandez Ruiperez, Alexander Polishchuk, Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes (arXiv:2008.00700)

The observation that the study of super-structures in mathematics is usefully regarded as taking place over the base topos on the site of super points has been made around 1984 in

• Albert Schwarz, On the definition of superspace, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37–42, (russian original pdf)

• Alexander Voronov, Maps of supermanifolds , Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 43–48

and in

• V. Molotkov, Infinite-dimensional $\mathbb{Z}_2^k$-supermanifolds, ICTP preprints, IC/84/183, 1984.

A summary/review is in the appendix of

• Anatoly Konechny, Albert Schwarz,

  On $(k \oplus l|q)$-dimensional supermanifolds in Julius Wess, V. Akulov (eds.) Supersymmetry and Quantum Field Theory (Dmitry Volkov memorial volume), Lecture Notes in Physics 509, Springer 1998 (arXiv:hep-th/9706003, doi:10.1007/BFb0105222)

  Theory of $(k \oplus l|q)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486

Generalization to supergeometry over an arbitrary commutative ring, in particular $p$-adic rings, is given in

• Albert Schwarz, I. Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)

A review of all this as geometry in the topos over the category of superpoints is in

• Christoph Sachse, A Categorical Formulation of Superalgebra and Supergeometry (arXiv:0802.4067)

Formulation in terms of synthetic differential supergeometry is in

• David Carchedi, Dmitry Roytenberg, On theories of superalgebras of differentiable functions, Theory and Applications of Categories, Vol. 28, 2013, No. 30, pp 1022-1098. (arxiv:1211.6134, TAC)

For many more references see at supermanifold.

Plenty of discussion of supergeometry with an eye towards supersymmetry in quantum field theory is in

• P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison, E. Witten (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

especially in the contribution

• Pierre Deligne, Daniel Freed, Supersolutions (arXiv:hep-th/9901094).

The appendix there

• Sign manifesto (pdf)

means to sort out various sign conventions of relevance.

Discussion of how supersymmetry is universally induced in higher category theory/homotopy theory by the free abelian ∞-group on the point – the sphere spectrum – is in

• Mikhail Kapranov, Categorification of supersymmetry and stable homotopy groups of spheres, April 2013 (abstract, video)

For more on this see at superalgebra.

Discussion related to G-structure and Killing spinors includes

• Dmitri Alekseevsky, Vicente Cortés, Chandrashekar Devchand, Uwe Semmelmann, Killing spinors are Killing vector fields in Riemannian Supergeometry (arXiv:dg-ga/9704002)

Discussion of classical field theory with fermions as taking place on supermanifolds is in the following references

• Pierre Deligne, Daniel Freed, Classical field theory (1999) (pdf)

  this is a chapter in

  P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison, E. Witten (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

• Daniel Freed, Classical field theory and Supersymmetry, IAS/Park City Mathematics Series Volume 11 (2001) (pdf)

• Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily, chapter 3 of Advanced classical field theory, World Scientific (2009)

• Gennadi Sardanashvily, Grassmann-graded Lagrangian theory of even and odd variables (arXiv:1206.2508)

• Gennadi Sardanashvily, Noether’s Theorems: Applications in Mechanics and Field Theory, Studies in Variational Geometry, 2016

Discussion on the context of super- string theory (especially regarding super Riemann surfaces):

• Eric D'Hoker, Duong Phong, Complex geometry and supergeometry, Current Developments in Mathematics 2005 (2007): 1-40 (euclid:1223654523)

Generalization of Artin's representability theorem to supergeometry:

• Nadia Ott, Artin’s theorems in supergeometry (arXiv:2110.12816)